An intersection formula for CM cycles on Lubin–Tate spaces

Abstract

We give an explicit formula for the arithmetic intersection number of complex multiplication (CM) cycles on Lubin–Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions $K_1,K_2$ over a non-Archimedean local field F. Our formula works for all cases: $K_1$ and $K_2$ can be either the same or different, ramified or unramified over F. This formula translates the linear arithmetic fundamental lemma (linear AFL) into a comparison of integrals. As an example, we prove the linear AFL for ${\mathrm{GL}}_2(F)$.